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added to periodic ring spectrum and to periodic cohomology theory a brief paragraph on looping/delooping periodicity on the $\infty$-modules, with a pointer to this MO discussion
What’s an interesting example of a 6-periodic ring spectrum?
And more generally for $(4k+2)$-periodicity, with $k \in \mathbb{N}$. Does anything spring to mind that would follow such a pattern in periodicity?
This reminds me of Qiaochu’s question here.
Thanks, Todd. Yes, I had seen that question when looking around yesterday.
Somehow what I am asking here should be more elementary. For instance where Qiaochu invokes the ABS orientation to explain 8-fold periodicity in manifolds from that of the $KO$-ring spectrum, here I’d be happy to just observe that $KO$ is 8-periodic. But I am really wondering about 6-periodicity.
I may be wrong, but recenty it occurred to me that the 2-periodicity in elliptic cohomology is to be thought of as related to the 2+-dimensionality of Chern-Simons theory. Because, as in this MO question there is naturally an elliptic line bundle in codimension 1 and in codimension 3, one being the point restriction of the other, and for the one in codimension 3 to have a chance to reproduce the one in codimenesion 1 by transgression, we need 2-periodicity of its fibers. If that’s the right perspective, then for 7d CS theory we’d expect to need 6-periodicity, and so that’s why I was wondering if there are any candidates for that.
In any case, of course one may alsways force anything to become $n$-periodic for any $n$. But I’d be happy to see ring spectra arising “in nature” that are 6-periodic.
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